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and "the direction of the force" must by no means be confounded together. If the former be known, the latter is necessarily known also; but if only the latter be given, the precise line in which the force produces, or tends to produce, motion, is a matter of uncertainty, and all that can be said respecting it is, that the line of action is some line or other parallel to the given line.]

(3) If when two or more forces are applied to a body, or at some point, and no motion is produced, they are said to counteract, or to balance one another, or to be in equilibrium.

11. Forces may be properly represented by lines.

Since lines may be drawn of any length and in any direction from a point, the lines in which forces act, and the ratios that the forces bear to one another, may be represented by drawing lines coinciding with the lines in which the forces act, and bearing to one another the same ratios that the forces themselves bear to one another.



[Among other advantages that attend the indicating in this way the magnitudes and directions of forces, the addition and subtraction of such forces as act at a point in the same straight line is easily effected. Thus if a certain force act at A in the line AH, and we take AB to represent it, and another force, half the size of the former, act at A in the same line, and also tend to move the body from A towards H, then by taking BC equal to the half of AB we get a line AC properly representing the whole pressure at A, both with Krespect to the magnitude of that pressure and the line in which it acts. And in like manner, if a force equal to half the original force AB act at 4 in the line AH, but tends to move the body at A from A towards K, half the pressure of the former force will be counteracted by this new force. Cutting off, therefore, from the line AB a part BD equal to the half of AB, the effective pressure still remaining will be properly represented by AD, with respect to both its magnitude and its line of action.]

12. [NOTE. It will be gathered from the last Article, that a force AB applied at A is not the same thing as a force BA applied at that point; for a force AB would tend to move a body at A in the line KH towards H, but a force BA would tend to move a body at A in the line KH towards K. It is not therefore indifferent whether "a force AB" or we say 66 a force BA"; since though the two forces represented by AB and BA are exactly the same in magnitude, and also act in the same straight line, yet they tend to produce motions that are directly opposite to one another; the force AB tending to move the body at A towards H, and the force BA tending to move the body at A towards K.]

13. [The effect produced by any force is the same at whatever point in its line of action it is applied.


For if a heavy body P be suspended by a string CB, the B force necessary to prevent P falling to the earth is the same whether it be applied at A, or B, or C;-the weight of the string being either neglected, or the weight of that portion of it which is supported along with the heavy body P, being counterbalanced, or otherwise accounted for.]



14. [DEF. If a string fastened at one end be pulled by a force applied at the other end, the resistance to motion made by the string at any point in it is called the TENSION of the string at that point.

If the string be supposed to be without weight, it will follow from Art. 13 that the tension at every point of it is the same,-namely the force by which it is pulled.]

15. To recapitulate the substance of this Chapter.

(1) MATTER is the substance of which all bodies are composed. It is found, by universal experience, to possess the property of endeavouring to move towards the earth.

(2) The precise amount of the exertion which any particular body makes to move towards the earth is called the WEIGHT of that body.

This WEIGHT of a body is measured by determining how much the tendency of the body to move towards the earth is greater than the tendency of some other given body, of a certain size and formed of a certain material, which is taken for a standard, and to which the name of a grain, an ounce, a pound, or a ton is given, as the case may be.

(3) THE QUANTITIES OF MATTER contained in different bodies (whether the bodies be great or small, rare like gas or dense like lead), is proportional, (not equal,)-to the weights of the bodies.

(4) The DENSITIES of different substances are proportional,-(not equal,)—to the weights of equal bulks of the substances.

(5) I. Whatever moves, or tends to move, matter existing under any form whatsoever, is called FORCE. FORCES that prevent motion taking place,—that is, STATICAL FORCES,—are measured by the number of pounds they would support if they acted vertically upwards.

II. THE LINE OF A FORCE'S ACTION is the line in which the force tends to produce or prevent motion.

III. THE DIRECTION OF A FORCE is indicated either by the line of its action, or by any line parallel to the line of its action.

IV. The magnitudes of forces, and the lines in which they act, may be represented by straight lines properly drawn.

V. If AB represent a force acting on a point A, and BA represent another force acting on the same point, the two forces AB and BA are equal, and tend to produce motion in the same straight line, but in opposite ways from the point A.

VI. To investigate the effects produced by a force we must have given,-1st, the magnitude of the force, which is known if we know how many pounds the force would support; 2nd, the point at which the force is applied; and 3rd, the direction in which it acts; (for knowing the direction of the force and the point it is applied at, the line in which the force acts may be determined by drawing a line through the given point parallel to the given direction).




16. DEFS. (1) A PLANE, or a PLANE SUPERFICIES, is that superficies, or surface, in which if any two points be taken the straight line joining them lies wholly within the superficies. (EUCLID I; Def. 7.)

(2) A SOLID is that which hath length, breadth, and thickness. (EUCLID XI; Def. 1.)

(3) PARALLEL PLANES are such as do not meet one another, however far they may be produced. (EUCLID XI; Def. 8.)

(4) A PRISM is a solid bounded by plane rectilineal figures, two of which,

which are parallel to each other, are similar and equal triangles squares or poly





gons, and all the remaining plane rectilineal figures that bound the prism are rectangles.

[Thus, let ABCD and abcd be two equal and similar quadrilateral figures, placed with their planes parallel, and let all the figures, (such as ABba,) that are formed by joining the equal angles of ABCD and abcd, be rectangles; then the solid included by these rectangles and by the ends, or bases, ABCD and abcd, is called a prism.

The length of the prism is any one of the edges Aa, Bb, &c., which lines, being the sides of adjacent rectangles, are all equal to one another.]

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